Mathematics

o Qualitative and quantitative analysis - Gives students a "feel" for the equations that enables them to explore a common-sense line of questioning and encourages students to think about dynamical systems.

o Consistent approach - Uses similar methods to solve  various systems (algebraic, differential, and iterative)  thus reinforcing concepts.

o Computer graphics analysis approach - Encourages a real understanding of nonlinear differential and iterative equations.

o Abundant illustrations and figures - Approximately three times as many figures as competing texts.

o Accompanying CD with IDE software — Includes 97 interactive illustrations that allow the student to experiment visually with differential equations which are linked to corresponding models and graphs.

 o Qualitative analysis of nonlinear systems. A vital extension of linear systems with a graphics approach that enables students to observe the effects of changing parameters and work with applications in other disciplines.

o Discrete dynamical systems - Provides a current and realistic introduction to the more complicated behaviors occurring in discrete iterative systems.

o Control theory - Enables instructors to introduce one of the most important applications of differential equations.

• Significant revision/rewriting throughout – Strengthens the link between linear algebra and differential equations, improves clarity, and responds to reader suggestions wherever possible.
• New sections on the matrix exponential and the Laplace transform.  
• Over 400 innovative new problems.
• The method of variation of parameters is moved to the more traditional position in the chapter on second-order equation (previously found in the chapter on systems).
• Enlarged sections on nonhomogeneous equations and Laplace transform, both for differential equations and their systems.

1 First-Order Differential Equations

1.1 Dynamical Systems: Modeling 1

1.2 Solutions and Direction Fields: Qualitative Analysis 11

1.3 Separation of Variables: Quantitative Analysis 25

1.4 Approximation Methods: Numerical Analysis 33

1.5 Picard’s Theorem: Theoretical Analysis 46

2.1 Linear Equations: The Nature of Their Solutions 55

2.2 Solving the First-Order Linear Differential Equation 63

2.3 Growth and Decay Phenomena 73

2.4 Linear Models: Mixing and Cooling 80

2.5 Nonlinear Models: Logistic Equation 87

2.6 Systems of Differential Equations: A First Look 100

3.1 Matrices: Sums and Products 115

3.2 Systems of Linear Equations 130

3.3 The Inverse of a Matrix 146

3.4 Determinants and Cramer’s Rule 156

3.5 Vector Spaces and Subspaces 167

3.6 Basis and Dimension 177

4.1 The Harmonic Oscillator 195

4.2 Real Characteristic Roots 210

4.3 Complex Characteristic Roots 229

4.4 Undetermined Coefficients 244

4.5 Variation of Parameters 255

4.6 Forced Oscillations 261

4.7 Conservation and Conversion 274

5.1 Linear Transformations 285

5.2 Properties of Linear Transformations 300

5.3 Eigenvalues and Eigenvectors 311

5.4 Coordinates and Diagonalization 327

6.1 Theory of Linear DE Systems 343

6.2 Linear Systems with Real Eigenvalues 357

6.3 Linear Systems with Nonreal Eigenvalues 372

6.4 Stability and Linear Classification 384

6.5 Decoupling a Linear DE System 394

6.6 Matrix Exponential 400

6.7 Nonhomogeneous Linear Systems 410

7.1 Nonlinear Systems 421

7.2 Linearization 431

7.3 Numerical Solutions 441

7.4 Chaos, Strange Attractors, and Period Doubling 449

7.5 Chaos in Forced Nonlinear Systems 456

8.1 The Laplace Transform and Its Inverse 467

8.2 Solving DEs and IVPs with Laplace Transforms 475

8.3 The Step Function and the Delta Function 485

8.4 The Convolution Integral and the Transfer Function 499

8.5 Laplace Transform Solution of Linear Systems 509

9.1 Iterative Equations 517

9.2 Linear Iterative Systems 530

9.3 Nonlinear Iterative Equations: Chaos Again 542

10.1 Feedback Controls 557

10.2 Introduction to Optimal Control 567

10.3 Pontryagin Maximum Principle 579

Differential Equations and Linear Algebra  (Mathematics)

Author's Web Site

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